Methods and systems for confining charged particles to a compact orbit during acceleration using a non-scaling fixed field alternating gradient magnetic field

ABSTRACT

A method is described wherein a beam of charged particles is confined to an orbit within a compact region of space as it is accelerated across a wide range of energies. This confinement is achieved using a non-scaling magnetic field based on the Fixed Alternating Gradient principle where the field strength includes non-linear components. Examples of magnet configurations designed using this method are disclosed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This present application claims priority to U.S. patent application Ser.No. 13/034,931 entitled “Methods And Systems For Confining ChargedParticles To A Compact Orbit During Acceleration Using A Non-ScalingFixed Field Alternating Gradient Magnetic Field” filed Feb. 25, 2011,and the benefit of U.S. Provisional Patent Application Ser. No.61/308,142 entitled “Methods And Systems For Confining Charged ParticlesTo A Compact Orbit During Acceleration Using A Non-Scaling Fixed FieldAlternating Gradient Magnetic Field” which was filed on Feb. 25, 2010 byWilliam Bertozzi, Wilbur Franklin, Carol Johnstone and Robert J. Ledoux,both of which are hereby incorporated herein by reference.

FIELD OF THE INVENTION

This application relates to the design of particle accelerators, and inparticular to non-scaling fixed field alternating gradient machines inwhich charged particles are confined to compact orbits while beingaccelerated to a desired energy.

BACKGROUND

Particle accelerators can be generally grouped into two basiccategories: accelerators which recirculate particles and those whichaccelerate particles in a line. The latter (linacs) require transversemagnetic fields only to focus or form a beam with finite spatial extent;the net transverse magnetic field at the center of the beam is zero. Thefirst category, however, necessitates the use of a system of guidemagnets to confine the beam in a circular or spiral orbit duringacceleration. This magnetic field can take many different forms. Forexample, in the case of a synchrotron, the magnetic field is varied withtime to keep charged particles confined to a closed orbit which lieswithin a very limited radial range as the particles accelerate and gainenergy; the beam orbit experiences no or little change in radius withenergy. Synchrotrons have small magnetic component apertures whichaccommodate the transverse size of the circulating beam but the magneticfield must be ramped, or pulsed, in synch with the beam energy.Synchrotrons therefore have limited duty cycle due to the time requiredto pulse and recycle a magnetic field. A betatron behaves in a similarfashion, requiring the guide field to vary with energy.

In a recirculating accelerator with fixed magnetic fields, no suchcycling is necessary, but the position of the particle beam changes asit gains energy. Stable orbits require the integrated magnetic strengthto scale with the momentum of the particles, and to accurately track theposition of the particle beam as it moves outward across the magneticaperture during acceleration. In the case of some cyclotrons, the magnetfield is very nearly uniform in space and time and particles move intolarger orbits as their energies rise, with a correspondingly longer pathlength in the field (and therefore increased integrated field strengthas a function of energy). In the case of a fixed field alternatinggradient (FFAG) accelerator, the magnetic field strength at a givenpoint in space does not vary with time, but its spatial variation withradius can be large in order to match the increasing momentum of theparticles as they gain energy and their orbits change accordingly; thatis, in order to limit the required increase in orbit radius, themagnetic field can increase sharply as a function of radius.

The use of a field profile with strong spatial variation can permitparticles of very different energies to coexist in close andpre-determined proximity. For applications requiring acceleration ofintense beams within a compact space, this is a desirable property.There are many situations where a small footprint or compact size isdesirable or necessary because of limitations in space and/orrequirements of portability. A small footprint can also be important toachieve high duty cycle and consequently increased beam intensity with amodest acceleration system. As a result, accelerators using the FFAGprinciple for the guide magnets have attracted particular attention incommercial and research applications requiring high beam power, highduty cycle, reliability, and precisely controlled beams at reasonablecost. See C. Prior, Editor, ICFA Beam Dynamics Newsletter #43, August2007, http://www-bd-fnal.gov\icfabd\ Newsletter43.pdf ; M. K. Craddock,“New Concepts in FFAG Design for Secondary Beam Facilities and OtherApplications”, PAC'05, Knoxville, Tenn., USA, 16-20 May 2005, p. 261;Machida, http://hadron.kek.jp/˜machida/mirror/misl/publications20090415.pdf; RACCAM project,http://lpsc.in2p3.fr/service_accelerateurs/raccam.htm; E. Keil,http://keil.home.cern.ch/keil/keil.bib; CONFORM project, U.K. (EMMA andPAMELA), http://www.conform.ac.uk; Proceedings of InternationalWorkshops on FFAG Accelerators (FFAG00-FFAG08).

FFAG magnet systems and the resulting accelerators can be divided intoscaling and non-scaling types. Scaling FFAGs are characterized bygeometrically similar orbits of increasing radius. The magnetic field,both in radial sector designs (Keith R. Symon, A Strong FocussingAccelerator with a DC Ring Magnet, MURA Notes, Aug. 13, 1954; and D. W.Kerst, K. R. Symon, L. J. Laslett, L. W. Jones, and K. M. Terwilliger,“Fixed field alternating particle accelerators”, CERN SymposiumProceedings, v.1, 1956, p. 366) and in spiral sector designs (D. W.Kerst, “Properties of an Intersecting-Beam Accelerating System”, CERNSymposium Proceedings, v.1, 1956, pp. 36-39), follows the laws

B∝r^(k)F(θ),

B∝r^(k)G(Ψ)

where r is the radius and k, the constant field index, is the “scaling”attribute. F(θ) and G(Ψ) are dependent on the chosen design. (In thespiral ridge design, the parameter Ψ is related to the physical angle,θ, in a well understood manner related to design details.) Scalingmachines are theoretically designed such that at all energies during theacceleration cycle the particle beam executes a fixed number of betatronoscillations about a reference orbit in the course of a revolution. Thiscondition, referred to as constant betatron tune, helps to insure thatthe beam can be accelerated without encountering the strong low-orderresonances which lead to beam blow-up and eventual loss from particleshitting accelerator walls. Scaling FFAG accelerators directlyincorporate high-order multipole fields to achieve this constant tuneand, in general, require complex magnet shapes, pole profiles and, inthe case of the spiral sector design, elaborate edge shaping.

The non-scaling FFAG was originally conceived as a means for the rapidacceleration of muon beams. This type of design did not attempt toachieve a constant betatron tune, employed only a linear radialdependence for the magnetic field strength (thereby achieving a largedynamic aperture) and aimed for beam stability only during a briefacceleration cycle lasting for tens of orbits and spanning a limitedenergy range. These muon or rapid acceleration designs utilizedrectangular, fixed-field magnets that combined steering and transversefocusing using only linear (quadrupole) gradients to monotonicallyincrease the field with radius. However, the non-uniform,energy-dependent betatron tunes resulted in the beam crossing manyresonances, thus making this approach untenable for gradual accelerationwhich requires beam stability over a much larger number of turns.

An innovative non-scaling approach to gradual acceleration wassubsequently proposed in which the constant tune feature wassuccessfully combined with the simplicity of linear-field, non-scalingFFAG components. (See US Published Patent Application 2007/0273383,“Tune-stabilized, non-scaling, fixed-field, alternating gradientaccelerator”, Johnstone, Carol J.) This non-scaling approach is termed alinear-field, linear-edge FFAG; in it, weak and strong focusingprinciples (both edge and linear-gradient focusing) are applied tofixed-field combined-function magnets, but now with canted entrance andexit faces, so as to stabilize machine tunes. This stabilization occursthough an appropriate increase in the net or integrated field withradius. However, this linear-field non-scaling FFAG remains limited interms of machine design and features. Generation of stable tunesrequires either the use of large component apertures incompatible with acompact system, or the imposition of a restricted acceleration range.While momentum gains of at approximately 600% have been achieved withthis type of design, the approach fails to provide the compactness andexpanded tune stability needed for applications requiring largermomentum gains with modest acceleration per turn.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows the layout and certain of the parameters of half of astandard unit cell of a pair of FFAG magnets. The figure consists ofhalf of a horizontally focusing (F) magnet on the left and half of ahorizontally defocusing (D) magnet on the right. Because the figuredisplays half of each magnet, it is reflected at either end to producethe full-cell unit.

FIG. 1B shows the layout and certain other parameters of the half of astandard unit cell of FFAG magnets shown in FIG. 1A.

FIG. 2 shows an entire FFAG magnet system as constructed from eightidentical unit cells as in FIGS. 1A and 1B to form a recirculating ringlayout for a 3-MeV machine. (The number of cells can vary depending onenergy, ring size, and magnet aperture.) The location of the inductioncore is shown schematically; its specific location with respect to thefocusing and defocussing magnets in any concrete design may varydepending on practical design considerations.

FIG. 3 shows the full ring layout of a 9-MeV compact electronaccelerator constructed from the unit cells of FIGS. 1A and 1B. The9-MeV design requires nine full cells instead of the eightcharacteristic of lower energy machines. The location of the inductioncore is shown schematically; its specific location with respect to thefocusing and defocussing magnets in any concrete design may varydepending on practical design considerations.

DETAILED DESCRIPTION OF EMBODIMENTS

In this disclosure we introduce FFAG designs that achieve compactnesswithin the context of the non-scaling approach by removing thelinear-field condition, as well as a method for designing suchaccelerators. The guide field magnets in these new designs retain thesimple wedge shapes, but nonlinear field components are systematicallyintroduced to realize more advanced machine properties and moredesirable designs than can be realized with linear fields. The type andmagnitude of the nonlinear field content remain dependent on machinegeometry, energy reach, and application.

The embodiments described herein are exemplary of possible applicationsof the technology disclosed herein for accelerating charged particlebeams. Those experienced in the art will recognize that there areextensions, modifications and other arrangements of the importantelements disclosed that can be implemented and they are included as partof this disclosure.

As examples of accelerators designed by the methods disclosed herein, wepresent four basic designs:

a) an electron accelerator from kinetic energy of 50 keV to 3 MeV,representing a momentum gain of 1500% ;

b) a 3.5 MeV ultra-compact version which has a significantly reducedfootprint relative to the first machine;

c) a 4 MeV ultra-compact version which also has a significantly reducedfootprint relative to the first machine; and

d) a more advanced version with an enhanced acceleration cycle fromkinetic energy of 50 keV to 9 MeV, representing a momentum gain of morethan 4000%.

These designs demonstrate the nature of the field behavior required toconstruct a non-scaling FFAG accommodating a momentum range as large asa factor of 41 between injection and extraction (a kinetic energy rangeas large as a factor of 180), within a restricted size, that can bephysically realized with good magnet design using familiar componentswhile employing minimal non-linear dynamics and field patterns.

In an induction machine the smallest inner dimensions are set mostly bythe area of the induction core that achieves the necessary flux changeto achieve the required momentum/energy gain (and as large as possibleduty cycle) with minimal power consumption. The extraction radius isestablished by the desire to minimize the size of the machine (and alsomaximize the duty cycle) using the magnetic materials at hand. As notedabove, the linear field non-scaling design cannot achieve these combinedgoals for an extraction kinetic energy between 3 and 9 MeV; the firstand fourth designs presented herein demonstrate the achievement of theseenergies. In the second and third designs, the induction core is reducedin size compared to the first design to accommodate the reduced radii.Energies of 3.5 and 4 MeV are possible at differing duty cycles. Theminimum required area of an induction core is determined by the energyto be gained by the acceleration. Induction core areas larger than thisminimum permit larger duty cycles up to a maximum of 50%.

Reduction in the size of the induction core allows a reduction in bothinjection radius and extraction radius. Again, the linear design cannotachieve these combined goals.

In the design method disclosed herein, which was used to develop thespecific accelerator designs also disclosed, the magnetic fieldexpansion, and in particular the nonlinear content thereof, isultimately the product of an optimization search to best fit specificdesign criteria. The optimized 3-MeV design exhibits significantlylower-order multipole magnetic field content than the 9-MeV design (or,alternatively, greater tune stability for the same multipole content).The 3-MeV design has an average injection radius of 20 cm and an averageextraction radius of 42 cm. In the 9 MeV machine, the injection radiuswas similarly constrained to be less than 24 cm and the extractionradius less than 44 cm. (In the 9 MeV machine design, the average radiusof the 3 MeV orbit is about 34 cm due to the more rapid field increaseas a function of radius as compared to the 3 MeV machine design.)

Acceleration to other energies with comparable momentum/energy gain,optionally with other differing design features as well, is an obviousextension of the concepts and design methods herein described. Thereduced-induction-core-size 3.5 and 4 MeV machines are two suchexamples, wherein other dimensions and parameters are optimized to takeadvantage of the reduced core size to reduce overall footprint andincrease duty cycle. For the machines described here, the magnets arepractical to manufacture with field strengths and profiles that allowconstruction with either electromagnets, or permanent magneticmaterials.

As set forth above, the initial approach to tune stabilization in alinear-field non-scaling FFAG was developed by C. Johnstone. (See USPublished Patent Application 2007/0273383, “Tune-stabilized,non-scaling, fixed-field, alternating gradient accelerator”, Johnstone,Carol J.; C. Johnstone, et. al., “Non-scaling FFAG Variants for HEP andMedical Applications,” to be published in Proceedings of the 2009Particle Accelerator Conference, Vancouver, Calif., May 4-8, 2009; C.Johnstone, et. al., “A New Nonscaling FFAG for Medical Applications,”ICFA Beam Dynamics Newsletter No. 43, July, 2007,http://www-bd.fnal.gov/icfabd/Newsletter43. pdf, pp. 125-132; C.Johnstone, et. al., “New Nonscaling FFAG for Medical Applications,”Proceedings of the 2007 Particle Accelerator Conference, Albuquerque, N.Mex., Jun. 25-29, 2007, pp. 2951; C. Johnstone, et. al.,“Tune-stabilized Linear Field FFAG for Carbon Therapy”, Proceedings ofthe 2006 European Particle Accelerator Conference, Edinburgh, UK, Jun.26-30, 2006, pp. 2290-2292) Her approach involved the use of ananalytical model which is also a starting point for the methods anddesigns described herein. In that model, the magnet system is dividedinto a set of identical cells, as shown in FIGS. 1A and 1B. FIGS. 2 and3 show exemplary complete ring layouts based upon such cells. Eachindividual cell in the rings in FIGS. 2 and 3 consists of two steeringmagnets with shaped edges, that also provide transverse envelopecontrol, namely an F-magnet for focusing the beam horizontally whiledefocusing it vertically, and a D-magnet which defocuses the beamhorizontally and focuses it vertically, as shown in FIGS. 1A and 1B.

Since the ring is entirely repetitive, it can be described dynamicallyby the properties of its constituent cells.

The following is a list of relevant terms and parameters for the ringmagnets:

P_(extract) Extraction momentum [MeV/c];P_(inject) Injection momentum [MeV/c];ρ_(ef) Radius of trajectory in F magnet at extraction [m];ρ_(ed) Radius of trajectory in D magnet at extraction [m];ρ_(if) Radius of trajectory in F magnet at injection [m];ρ_(id) Radius of trajectory in D magnet at injection [m];η_(if) Edge angle of F magnet at injection [rad];ηhd ef Edge angle of F magnet at extraction [rad];

As currently modeled, η_(ef)=η_(if)=η_(f) , the edge angle relative tothe sector angle at extraction as shown in FIG. 1A.

η_(id) Edge angle of D magnet at injection [rad];

η_(ed) Edge angle of D magnet at extraction [rad];

As currently modeled and as in the F magnet, both of these edge anglesare set to be equal. However, these angles for the D magnet are defineddifferently as shown in FIG. 1A. This edge angle is defined relative tothe line through the center of the F magnet.

D_(e) Drift length between F and D magnets at extraction [m];

D_(i) Drift length between F and D magnets at injection [m];

L_(ef) Trajectory length in half of the F magnet at extraction [m];L_(if) Trajectory length in half of the F magnet at injection [m];L_(ed) Trajectory length in half of the D magnet at extraction [m];L_(id) Trajectory length in half of the D magnet at injection [m];δx_(if) Distance from injection orbit to extraction orbit, center of Fmagnet [m];δx_(id) Distance from injection orbit to extraction orbit, center of Dmagnet [m];θ_(ef) Angle of trajectory in half F magnet at extraction [rad];θ_(ed) Angle of trajectory in half D magnet at extraction [rad];θ_(if) Angle of trajectory in half F magnet at injection [rad];θ_(id) Angle of trajectory in half D magnet at injection [rad];θ_(id) Angle of trajectory in half D magnet at injection [rad];k_(ef)=30 B′_(ef)/P_(extract)k_(ef)=30 B′_(ef)/P_(extract) Normalizedfocusing strength of F magnet at extraction [m⁻²] (defined in terms ofthe local field derivative);k_(ed)=B′_(ed)/P_(extract)k_(ef)=30 B′_(ef)/P_(extract)k_(ed)=30B′_(ed)/P_(extract) Normalized focusing strength of D magnet atextraction [m⁻²] (defined in terms of local field derivative);k_(if)=30 B′_(if)/P_(inject)k_(ef)=30 B′_(ef)/P_(extract)k_(if)=30B′_(if)/P_(inject) Normalized focusing strength of F magnet at injection[m⁻²] (defined in terms of local field derivative);k_(id)=30 B′_(id)/P_(inject)k_(ef)=30 B′_(ef)/P_(extract)k_(id)=30B′_(id)/P_(inject) Normalized focusing strength of D magnet at injection[m⁻² ] (defined in terms of local field derivative);L_(ehalf)=L_(ef)+L_(ed)+D_(e) Half-cell trajectory length at extraction[m]; andL_(ehalf)=L_(if)+L_(id)+D_(i) Half-cell trajectory length at injection[m].

The following identities may be used to relate these variables to basicmachine parameters that in turn may be used in equations:

θ_(ef) =L _(ef)/ρ_(ef)   1)

θ_(ed) =L _(ed)/ρ_(ed)   2)

θ_(if) =L _(if)/ρ_(if)   3)

θ_(id) =L _(id)/ρ_(id)   4)

ρ_(ef) =P _(extract)/30B _(ef)   5)

ρ_(ed) =P _(extract)/30B _(ed)   6)

ρ_(if) =P _(inject)/30B _(if)   7)

ρ_(id) =P _(inject)/30B _(id)   8)

9) Magnetic field expansion at extraction in F magnet:

B _(ef) =B _(0f) +a _(f) δx _(ef) +b _(f) δx _(ef) ² +c _(f) δx _(ef) ³d _(f) δx _(ef) ⁴ +e _(f) δx _(ef) ⁵ +f _(f) δx _(ef) ⁶

10) Magnetic field expansion at extraction in D magnet:

B _(ed) =B _(0d) +a _(d) δx _(ed) +b _(d) δx _(ed) ² +c _(d) δx _(ed) ³+d _(d) δx _(ed) ⁴ +e _(d) δx _(ed) ⁵ +f _(d) δx _(ed) ⁶

11) Magnetic field expansion at injection in F magnet:

B _(if) =B _(0f) +a _(f)(δx _(ef) −δx _(if))+b _(f)(δx _(ef) −δx _(if))²+c _(f)(δx _(ef) −δx _(if))³ ++d _(f)(δx _(ef) −δx _(if))⁴ e _(f)(δx_(ef) −δx _(if))⁵ +f _(f)(δx _(ef) −δx _(if))⁶

12) Magnetic field expansion at injection in D magnet:

B _(id) =b _(0d) +a _(d)(δx _(ed) −δx _(id))+b _(d)(δx _(ed) −δx _(id))²+c _(d)(δx _(ed) −δx _(id))³ ++d _(d)(δx _(ed) −δx _(id))⁴ +e _(d)(δx_(ed) −δx _(id))⁵ +f _(d)(δx _(ed) −δx _(id))⁶

Note that fields are expressed in kG and that the field expansion is invariables relative to the extraction orbit—for example, δx_(if) is thedistance from injection to extraction in the F magnet so that theincreasing values for the field correspond to increasing values ofradius. The value δx_(ef) is the distance from where the field has avalue B_(0f) to the extraction orbit. The point in radius where thefield has the value B_(0f) is the position about which the fieldexpansion is made. This position, along with the variable B_(0f) , isselected by the optimizer. (The extraction orbit proves to be the mostcritical in many designs and it also has the highest field values so itis often used as the fixed reference in these machine designs.) Asdiscussed more fully below, not all of the field expansion coefficientsset forth above are required in all of the machine designs—this justindicates the highest order used to date.

The properties of each individual cell may be described by sevenconstraint equations, discussed below, and twelve free parameters, alsodescribed below. The seven equations describe the reference trajectoriesand the “local” linear dynamics of beams at injection and extractionexplicitly expressed in terms of the physical attributes of the magneticcomponents: their radial field profiles and lengths. The magnets havelinear edge contours and the hard-edge model is implicitly assumed. Themethods described herein use these equations, with appropriatemodifications, extensions and additions to be discussed, for an initialoptimization/parameter search to find potential parameters for desireddesigns. (After doing so, the parameters derived from the hard edgemodel are inserted into a model with realistic edge magnetic fields andparticles in a beam are tracked to verify the stability of the potentialdesign.)

All parameters in these seven constraint equations can be expressed interms of twelve variables described below and illustrated in FIGS. 1Aand 1B, for the complete ring layout examples in FIGS. 2 and 3. In theseequations, “e” and “i” denote extraction and injection, subscripts “f”and “d”, horizontally focusing and defocusing magnets, and “f” the thinlens focal length. The twelve variables are:

 1. D_(e) Drift distance between F and D magnets at extraction 2.-5.L_(if), L_(ef), L_(id), L_(ed) F and D Magnet half-lengths at injectionand extraction 6.-9. B_(if), B_(ef), B_(id), B_(ed) F and D Magnetfields at injection and extraction 10. δx_(if), Distance from theinjection orbit to the extraction orbit in the F magnet 11.-12. η_(f),and η_(d) Linear edge angles for the F and D-magnets.

The first four equations characterize horizontal and vertical tunerequirements at injection and extraction. Thin-lens versions of thefirst four equations are presented below for purposes of demonstration.For the initial design and choice of parameters, however, they arereplaced by the actual thick-lens equations to arrive at an initialdesign which has a stable tune; i.e. the first four equations are thetraces of the thick-lens linear matrices representing a full cell whichare the tunes across the cell. For a completely periodic lattice, thefirst four equations serve to specify the machine tune at injection andextraction (when multiplied by twice the number of cells). In the thinlens representation the focal length is related to the half cell tune by

${\sin \frac{\phi}{2}} = {{{L_{half}/f}\; {\sin \left( {\phi/2} \right)}} = {L_{half}/f}}$

where L_(half) is the half-cell length and φ is the full cell phaseadvance. For example, for φ=90°, f=1.4 L_(half) where L_(half) is thesum of the two magnet half-lengths plus the intervening drift.) Anegative p reverses the sign of the edge crossing term in the verticalequation. The edge-angle convention here is opposite many conventionalusages—in this work a larger wedge angle implies an increasing value ofη.

As noted above, these seven constraint equations rely on the fact thatthe FFAGs here are completely periodic lattices where the base symmetrymodule is a half cell since there is reflection symmetry; i.e. thelattice is built from a half-cell plus a reflected half cell. Thehalf-cell symmetry—center of F magnet to center of D magnet—represents avery powerful constraint in that all derivatives are required to be nullat these points to enforce reflective symmetry and identical beginningand end optical functions for the full unit cell. With half cellsymmetry, the null derivatives further imply that reference orbits mustbe parallel at these points. Periodicity and parallel reference orbitsrequire that there be the same net bend per half cell.

This imposition of geometric closure of reference orbits is enforced inthe fifth equation (set forth below), which relates the net bend percell for injection and extraction trajectories.

The last two equations set forth below are also geometric in nature,linking magnetic lengths and drifts at injection to extraction throughthe sector angle, edge angle and alignment of the magnetic components,defining the physical linear edge and extent of the magnet. Theseequations govern the magnetic lengths at extraction and injectionconnected via the linear edge contour.

FIGS. 1A and 1B show the relation of the parameters in the sevenequations to the F and D physical magnet design.

The first five equations are general. They are derived from lineardynamics. That is, they exclude the angle term in the dynamics of theparticle Hamiltonian and also do not take into account the extent of thefringe fields. For the last two equations, particle dynamics isapproximated by impulses given at the center of the magnets.

The system of equations using the thin-lens approximations set forthbelow may be solved for magnetic designs assuming the approximationsmentioned above. However, as discussed above for the actual magneticdesign the four thin-lens equations are not used, but rather the processinvolves using the traces of the corresponding thick lens linearmatrices which represent the tune for the guide magnets under the sameassumptions—neglecting the angle term in the Hamiltonian and extent ofthe fringe fields. If the design is not stable, further designiterations using the optimizer may take place until a satisfactorydesign is reached.

The seven equations are as follows, showing the four thin-lens equationsfor simplicity in demonstration rather than the thick-lens equationsactually used. In the equations, f is the thin-lens focal length:

${\left. {{{\left. {{{{\left. {{\left. {{{{{\left. {{\left. {{{{{\left. \mspace{20mu} 1 \right)\mspace{14mu} k_{if}L_{if}} + \frac{\theta_{if}}{\rho_{if}} + \frac{\left( {\left( {\theta_{ef} - \theta_{if}} \right) + \eta_{if}} \right)}{\rho_{if}}} = \frac{1}{f_{if}}};}\mspace{20mu} 2} \right)\mspace{14mu} \text{?}}\mspace{20mu} 3} \right)\mspace{14mu} k_{ef}L_{ef}} + \frac{\theta_{ef}}{\rho_{ef}} + \frac{\eta_{ef}}{\rho_{ef}}} = \frac{1}{f_{ef}}};}\mspace{20mu} 4} \right)\mspace{14mu} \text{?}}\mspace{20mu} 5} \right)\mspace{14mu} \theta_{halfcell}} = {{\theta_{if} + \theta_{id}} = {\theta_{ef} + \theta_{ed}}}};}6} \right)\mspace{14mu} {L_{if}\left\lbrack {{\cos \left( \theta_{if} \right)} + {{\sin \left( \theta_{if} \right)}{\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}} \right\rbrack}} = {{L_{ef}{\cos \left( \theta_{ef} \right)}} - {\left\lbrack {{\delta \; x_{if}} - {L_{ef}{\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}}}7} \right)\mspace{14mu} L_{ed}{\cos \left( \theta_{ef} \right)}} = {{L_{id}{\cos \left( \theta_{if} \right)}} + {\delta \; x_{id}{\sin \left( {\theta_{id} + \theta_{if}} \right)}} + {\left\lbrack {{\delta \; x_{if}} + {\left( {L_{if} + D_{i}} \right){\sin \left( \theta_{if} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}$?indicates text missing or illegible when filed

As discussed above, only the linear gradient case—a monotonic linearincrease (or decrease) in field with distance from injection toextraction—was considered in the initial Johnstone work on non-scalingaccelerators referenced above. That case requires five technical choicesof parameters to fully solve these seven equations with twelveparameters; generally, the five may be chosen to be the distance betweenextraction and injection in the F-magnet, the drift length between themagnets at extraction, and 3 injection/extraction fields between theF-magnet and D -magnet combined-function magnets. With only lineargradients involved, the solutions are wedge-shaped, off-centerquadrupoles.

However, when the initial design solutions obtained in the above mannerare tested for tune stability, as discussed above, it is found that verycompact, stable orbits cannot be achieved using this linear-fieldnon-scaling approach for acceleration from 50 keV to the energy range 3to 9 MeV, within the compact size framework described above (20 to 44 cmradii). Furthermore, modifying the seven equations described above toaccommodate curved rather than linear magnetic field boundaries alsofails to produce practical magnetic component and lattice designs overthe full acceleration range from 50 keV to either 3 or 9 MeV.

Rather, studies show that relaxing the linear field gradient constraintand including higher field components in the non-scaling frameworkpermits achieving compact designs with practical magnet technologies,for acceleration ranges of the magnitude set forth above where thelinear field gradient solution fails. In these studies the linear radialfield profile was replaced with a high-order field expansion in thetrajectory and dynamical equations.

Dynamical equations at intermediate energies—identical to thoseimplemented at injection and extraction—could be added to allow for theadditional degrees of freedom introduced by the nonlinear fieldcomponents, but finding exact solutions with such nonlinear equations isimpractical at best.

Instead, practical starting solutions were discovered using nativeoptimizers within Mathematica® to explore the “local” parameter space.In this approach, technically infeasible designs are eliminated bysetting limits on the search parameters to exclude nonphysical magnetlengths and unachievable field strengths. In particular, magnet lengthsless than 0.01 m were excluded internally in the optimizer search andabsolute field strengths greater than 5 kG were not considered, as thisrepresents the effective permanent magnet field limit Other parameters,such as the magnet spacing at the extraction energy, were chosen basedupon technical considerations, namely cross-talk problems between themagnets and the need to accommodate coils, flanges, and vacuumconnections. In particular, the minimum magnet spacing at the extractionenergy was set to 0.05 m in the designs described herein, since this isconsidered to be the smallest feasible magnet spacing, or drift,considering both the technical considerations listed above andconventional extraction of the beam For significantly higher energies,or acceleration of heavier particles, this drift will increase, and, inthe latter case, the increase can be an order of magnitude or more.Drifts at injection and intermediate energies were constrained to belarger than at least 0.01 m for the same technical reasons. Theoptimizer techniques used are particularly powerful as the number ofconstraint equations used can be more or less than the number of searchparameters (which is not feasible when solving for an exact solution ofthe equations).

Once optimizers were implemented in the design process, additionalmachine-specific design criteria were added to further narrow thesolution set. These included constraints on the average radii atextraction and injection energies:

Inj Radius=Radi=NSector Lihalf/π  8)

Ext Radius=Rade=NSector Lehalf/π  9)

These equations for injection and extraction radius, which weresufficient when combined with the other constraints discussed herein topermit solutions to be obtained in the linear field gradient case, wereinsufficient for the non-linear gradient designs disclosed herein; asdiscussed further below, the radius for an additional energy wasrequired to be specified for the nonlinear 3, 3.5, 4, and 9-MeV designs.

In addition, a constraint was added in the bend per cell at injection toreflect the requirement that there be an integer number of magnetsectors. (This automatically extends to extraction because the bend atinjection is required to be equal to the bend at extraction in theoriginal seven global constraint equations).

${\left. 10 \right)\mspace{14mu} \frac{\pi}{NSector}} = {\theta_{id} + \theta_{if}}$

The number of sectors, constrained to be an integer via the bend/cellconstraint, was an important search parameter in the optimizationprocess. Typically, it was allowed to vary between 6 and 11. Forsolutions of approximately 3-MeV extraction energy, the optimal solutiontended to be 8 cells, while for higher energies such as 9 MeV the numberincreased to 9. Increasing the number of orbits within a fixed radialextent generally tends to require a higher number of sectors to reducebeam excursion.

Subordinate equations were also included in the optimizer equation list.The subordinate equations reflect the fact that the aperture in the Dmagnet and the drift at injection can be derived in terms of the basicparameters as defined in the seven primary equations from simplegeometric considerations (bend angles and lengths at injection andextraction and the magnet edge angles). These secondary, subordinateequations (set forth below) describe the dependent variables D_(i) andδx_(id) in terms of the independent variables given above, and are theinjection drift and orbit excursion in the D quadrupole betweeninjection and extraction, respectively.

δx _(id)cos(θ_(id)+θ_(if))=δx _(if)+Lihalf sin(θ_(if))−Lehalfsin(θ_(ef))   11)

D _(i)[cos(θ_(if))−sin(θ_(if))tan(η_(ed))]=(L _(ef) +D_(e))cos(θ_(ef))−L _(if)cos(θ_(if))+[δx _(if) +L _(if)sin(θ_(if))−(L_(ef) +D _(e))sin(θ_(ef))]tan(η_(ed))   12)

As stated above, once the linear field gradient constraint was relaxed,it was not possible to derive accelerator designs by fixing only theenergies at injection and extraction; rather, three energies—injection,extraction and an intermediate energy—were found empirically to beoptimal for design for these non-scaling accelerators. Additionalintermediate energies, beyond one, although considered, served only tooverload and inhibit optimizer convergence in the highly nonlineardesigns considered here. (Although the optimizer was sensitive to theintermediate energy choice and yielded the designs disclosed herein whenthe energy was chosen to be close to the injection energy, at 356 keV,other intermediate values can also be used in this design method.)

In total, twenty equations were eventually utilized in the optimizerleast squares merit function to describe the injection energy, theextraction energy, and the intermediate energy. These twenty finalequations include the twelve equations listed above and eight additionalequations relating to the intermediate energy specified:

${\left. {{{\left. {{{\left. {{{\left. {{{\left. {{{{{\left. {{{{{\left. \mspace{20mu} 13 \right)\mspace{14mu} k_{3\; f}L_{3\; f}} + \frac{\theta_{3\; f}}{\rho_{3\; f}} + \frac{\left( {\left( {\theta_{ef} - \theta_{3\; f}} \right) + \eta_{if}} \right)}{\rho_{3\; f}}} = \frac{1}{f_{3\; f}}};}\mspace{20mu} 14} \right)\mspace{14mu} k_{3\; d}L_{3\; d}} + \frac{\theta_{3\; f} + \eta_{3\; d}}{\rho_{3\; d}}} = \frac{1}{f_{3\; d}}};}\; \mspace{20mu} 15} \right)\mspace{14mu} \theta_{halfcell}} = {{\theta_{if} + \theta_{id}} = {\theta_{3\; f} + \theta_{3\; d}}}}16} \right)\mspace{14mu} {L_{3\; f}\left\lbrack {{\cos \left( \theta_{3\; f} \right)} + {{\sin \left( \theta_{3\; f} \right)}{\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}} \right\rbrack}} = {{{L_{ef}{\cos \left( \theta_{ef} \right)}} - {\left. \quad{{\left\lbrack {{\delta \; x_{3\; f}} - {L_{ef}{\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( {\theta_{ef} + \eta_{ef}} \right)}};17} \right)\mspace{14mu} L_{ed}{\cos \left( \theta_{ef} \right)}}} = {{L_{3\; d}{\cos \left( \theta_{3\; f} \right)}} + {\delta \; x_{3\; d}{\sin \left( {\theta_{3\; d} + \theta_{3\; f}} \right)}} + {\left\lbrack {{\delta \; x_{3\; f}} + {\left( {L_{3\; f} + D_{3}} \right){\sin \left( \theta_{3\; {if}} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}}}\mspace{20mu} 18} \right)\mspace{14mu} \delta \; x_{3\; d}{\cos \left( {\theta_{3\; d} + \theta_{3\; f}} \right)}} = {{\delta \; x_{3\; f}} + {L\; 3\; {half}\; {\sin \left( \theta_{3\; f} \right)}} - {{Lehalf}\; {\sin \left( \theta_{ef} \right)}}}}19} \right)\mspace{14mu} {D_{3}\left\lbrack {{\cos \left( \theta_{3\; f} \right)} - {{\sin \left( \theta_{3\; f} \right)}{\tan \left( \eta_{ed} \right)}}} \right\rbrack}} = {{\left( {L_{ef} + D_{e}} \right){\cos \left( \theta_{ef} \right)}} - {L_{3\; f}{\cos \left( \theta_{3\; f} \right)}} + {\left\lbrack {{\delta \; x_{3\; f}} + {L_{3\; f}{\sin \left( \theta_{3\; f} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}}\mspace{20mu} 20} \right)\mspace{14mu} {Rad}\; 3} = \left. {N\; {sector}\; L\; 3\; {{half}/\pi}}\mspace{70mu} \middle| {{\delta \; x_{3\; f}} + \left( {L_{ef} + D_{e}} \right)} \right.$

These twenty equations include:

a) 6 horizontal and vertical tune constraints at the three energies (Eq.1-4 and 13-14);

b) 2 equations requiring equality in the bend/cell for all energies (Eq.5 and 15);

c) 4 equations connecting the magnet lengths between energies throughthe linear magnet edge contour (Eq. 6-7 and 16-17);

d) 1 constraint requiring that the total number of cells have an integervalue (Eq. 10);

e) 3 equations governing the desired average orbit radius at the threeenergies (Eq. 8-9 and 20);

f) 2 subordinate equations for the drift at energies less than theextraction energy (However, the drift at extraction is fixed at 5 cm toallow physical space for extraction (Eq. 11 and 18); and

g) 2 subordinate equations for the excursion of the orbit relative tothe extraction orbit in the D magnet (Eq. 12 and 19).

The optimizer is extremely sensitive to the parameter setup and limitsHowever, once an approximate solution is found with a merit functionthat is sufficiently small a machine is fully described. Using theseparameters the exact orbits at all momenta can be calculated and thetunes can be checked for stability over the entire range. If the tunesare not sufficiently stable as a function of energy the optimizer mustbe adjusted and another search performed.

3-MeV Machine Design

A 50 keV to 3 MeV (kinetic energy) compact accelerator design wasaccomplished within a 20 to 42 cm average radius, injection toextraction, using the design approach described and Mathematica® for theoptimization search.

For the 3 MeV design, as noted, three field points, injection (50 keVkinetic energy), extraction (3 MeV) and an intermediate energy (356keV), were used in the optimization. Since the coefficients describing afield expansion can be found exactly if the number of field points isequal to the order of the field plus 1, the three field points weresufficient to determine coefficients up to sextupole explicitly, andthis was carried out. In this design, it was found that the sextupolefield (without higher order multipole components) was sufficient tocontain the tune variation and permit beam stability. For this specificelectron machine design, therefore, the optimization search includedthree fields: injection, extraction, and an intermediate energy, withlimits implemented as to acceptable field values. The radial fieldexpansion coefficients up to the sextupole field were derived in termsof these three fields.

The following were the equations used to solve for coefficients of thefield expansion in terms of the individual field points.

B_(f) = B_(0 f) + a_(f)δ x + b_(f)δ x²$b_{f} = \frac{{\left( {B_{ef} - B_{if}} \right)\delta \; x_{3\; f}} - {\left( {B_{ef} - B_{3\; f}} \right)\delta \; x_{if}}}{\delta \; x_{if}\delta \; {x_{3\; f}\left( {{\delta \; x_{3\; f}} - {\delta \; x_{if}}} \right)}}$$a_{f} = \frac{\left\lbrack {\left( {B_{ef} - B_{3\; f}} \right) + {b_{f}\left( {{{- 2}\delta \; x_{ef}\delta \; x_{3\; f}} + {\delta \; x_{3\; f}^{2}}} \right)}} \right\rbrack}{\delta \; x_{3\; f}}$B_(0 f) = B_(ef) − a_(f)δ x_(ef) − b_(f)δ x_(ef)²

General parameters of the 20-42 cm 3-MeV machine are set forth inTable 1. As indicated, tune stabilization was accomplished with only a2^(nd) order field expansion:

F magnet: B(r)=0.14−2.37r+10.44r² and

D magnet: B(r)=0.07−1.15r−0.83r²

with all fields given in units of kG and r in cm. In this notation, theradius parameter, r, is referenced to the center of the machine. Thezero-field point in the radial profile is outside the physical magnetand does not correspond to the “intersection” of the edge contours as ina conventional radial-sector FFAG.

TABLE 1 General Parameters of the 3 MeV non-scaling FFAG, lineargradient plus sextupole field profile. Parameter Unit InjectionExtraction Energy Range MeV 0.050 3.0 Tune/cell (ν_(x) /ν_(y)) 2π-rad0.260/0.253 0.263/0.263 Average Radius m 0.198 0.417 Number of cells 8Straight section length m 0.029 0.052 (drift length) Peak Field kG F0.088 1.002 D −0.177 −0.491 Magnet Lengths m F 0.0877 0.1348 D 0.00940.0976 Apertures m F 0.218 D 0.179

In contrast, a linear field gradient would give an optimized fieldprofile of

F magnet: B(r)=−0.77+4.81 r and

D magnet: B(r)=0.26−1.39 r

where the constant term is actually the field at injection. Asdiscussed, the tune for such a linear field gradient design is much lessstable, Δv_(x)=0.094 and Δv_(y)=0.067 over the acceleration range, thanin the design with the nonlinear sextupole term, where Δv_(x)=0.009 andΔv_(y)=0.02 (note these are hard-edge Mathematica® results).

4 MeV and 3.5 MeV Ultra-Compact Machine Designs

The inner radius of the 3 MeV machine can be made smaller within theinduction concept because the induction core can be reduced and stillachieve the flux change required for 3 MeV. As explained earlier, theduty cycle will be concomitantly affected. To illustrate this point, amachine of 4 MeV was designed, resulting in the design in Table 2. The 4MeV design achieves an average injection orbit radius of 16.2 cm and anextraction radius of 35 cm. The intermediate point used for the designoptimization remained 0.7 MeV/c (356 keV).

Further reduction in size was achieved with a 3.5 MeV kinetic energymachine, to average radii for injection and extraction respectively ofapproximately 10 cm and 27 cm, which shows how efficient this approachis in achieving ultra-compact machine designs. The 3.5 MeV design isshown in Table 3.

To achieve this further compactness and tune stability required fieldorders higher than the sextupole field used in the 42 cm, 3-MeV machinedescribed above. (That is, use of a sextupole field expansion did notyield a design which demonstrated satisfactory beam stability.) To carrythis out, the fields in the dynamical equations are described in termsof a Taylor expansion, with the coefficients of the Taylor expansionadded to the optimizer parameter search array. Convergence of theoptimization using the Taylor expansion is considered successful when astable tune is produced over the entire acceleration range.

The successful field expansions for these respective machines are asfollows. The 4 MeV design required an F magnet field of:

B(r)=−0.045−0.17r ²+26.35r ³

and the D magnet field is given by

B(r)=0.40−20.74r ²+9.71r ³.

The ultra-compact 3.5 MeV design required an F magnet field of:

B(r)=0.20+36.52r ²−52.53r ³

and the D magnet field is given by

B(r)=−0.09+2.03+12.97r ²+14.11r ³.

Interestingly, the field content of the ultra-compact 3.5 MeV machinedoes not reduce to the same orders in the F and D magnets, and isdifferent from the larger-radius 4-MeV version.

It should be noted that direct extension to octupole, or a third-orderfield expansion, in the optimization technique, by adding a further setof equations for an additional intermediate energy, greatly complicatedcoefficient derivation, requiring four field specifications at fourenergies. This increase in the number of constraint equations (to 28)and enhancement of parameter space was found not to be tractable by theoptimizer, generating either unusable solutions or non-convergence.Therefore, the higher energy and more compact designs discussed here,which required multipoles higher than sextupole, were defined asdescribed above in the optimizer in terms of field expansioncoefficients.

TABLE 2 General parameters of the 4 MeV non-scaling FFAG designParameter Unit Injection Extraction Energy Range MeV 0.050 4 Tune/cell(ν_(x) /ν_(y)) 2π-rad 0.224/0.164 0.231/0.210 Average Radius m 0.1620.350 No. cells 8 Straight section length m 0.010 0.050 (drift length)Peak Field kG F 0.082 1.105 D −0.110 −1.488 Magnet Lengths m F 0.09490.1458 D 0.0134 0.0294 Apertures m F 0.190 D 0.162

TABLE 3 General parameters of the ultra compact 3.5 MeV non-scaling FFAGdesign Parameter Unit Injection Extraction Energy Range (kinetic energy)MeV 0.050 3.5 Tune/cell (ν_(x) /ν_(y)) 2π-rad 0.275/0.259 0.209/0.190Average Radius m 0.104 0.274 No. cells 8 Straight section length m 0.0160.047 (drift length) Peak Field kG F 0.500 2.034 D −0.000 −0.209 MagnetLengths m F 0.0112 0.061 D 0.0390 0.0598 Apertures m F 0.190 D 0.161

9-MeV Machine Design

As noted above, for energies above the 3 MeV design, field componentsabove the sextupole were required to obtain beam stability, and hence inaddition to using three field points (injection, extraction and anintermediate energy of 356 keV) to define equations for theoptimization, radial field expansion coefficients needed to besubsequently derived from the results of the optimizer search. Thisapproach was also used for the 9-MeV design, but the 9-MeV machineproved more challenging and the order of the field expansion used by theoptimizer increased rapidly; i.e. the field had to rise more rapidly viaa higher-order Taylor expansion to confine the outer orbits, with theirincreased energy, to within the required (average) 44-cm radius. (It wasnot feasible, as discussed above, to use multiple specific field pointsand so the expansion coefficients were fit directly in the optimizerimplementation used for the 9-MeV machine design.)

The difficulty in the 9 MeV and other higher energy cases arose from theincreasing complexity of the higher-order derivations for coefficientsin terms of field values. Even the derivation for coefficients up toonly octupole, which requires 4 field points, rapidly expanded into verylengthy equations, particularly for the r³ term. The field expansionitself proved more tractable for the optimizer. Thus, the higher energyprocedure is essentially the inverse of the procedure used in the 3 MeVdesigns: instead of fitting specific field values for specific differentenergies, which then are used to determine a unique set of coefficientsfor the field expansion, the individual coefficients of the Taylor fieldexpansion were optimized. The specific coefficient solutions, if theoptimizer search was successful, then generated field profilesassociated with orbits that fulfilled all the criteria (at the threeenergies) such as the desired extraction radius. (Limits can still beset on the field values generated from the expansion and the radii ofthe orbits in the optimizer setup.)

Although the field expansion found by the optimizer was very high fortune stabilization (using an r⁶ term), it was discovered that a strongcancellation exists in the fields between the quadrupole (linear) termand terms higher than octupole. As a result, these machines are thefirst known accelerators to initiate strong focusing starting withsextupole fields. Also notably, the field expansion truncateseffectively at the octupole term and inclusion of complex higher-orderfields is not required.

The F magnet field may be written as

B(r)=0.11−8.67r ²+41.93r ³

and the D magnet field is given by

B(r)=−0.03−1.26r ² −54.67il r ³.

Since the 9 MeV accelerator is a 9-cell ring as compared with the 8 cellrings at lower energies, the ring layout is shown in FIG. 3. Again thenumber of sectors is one of the search parameters and generallyincreases as a larger and larger energy range is required within thesame magnet aperture.

TABLE 4 General parameters of the 9 MeV non-scaling FFAG designParameter Unit Injection Extraction Energy Range (kinetic) MeV 0.050 9.Tune/cell (ν_(x) /ν_(y)) 2π-rad 0.233/0.236 0.241/0.227 Average Radius m0.206 0.437 No. cells 9 Straight section length m 0.028 0.050 (driftlength) Peak Field kG F 0.114 1.986 D −0.107 −2.811 Magnet Lengths m F0.0670 0.1650 D 0.0208 0.0398 Apertures m F 0.230 D 0.199

While the systems and methods disclosed herein have been particularlyshown and described with references to exemplary embodiments thereof,they are not so limited and it will be understood by those skilled inthe art that various changes in form and details may be made thereinwithout departing from the scope of the disclosure. It should berealized this invention is also capable of a wide variety of further andother embodiments within the scope of the invention. Those skilled inthe art will recognize or be able to ascertain using no more thanroutine experimentation, many equivalents to the exemplary embodimentsdescribed specifically herein. Such equivalents are intended to beencompassed in the scope of the present disclosure.

1. A non-scaling fixed field alternating-gradient particle accelerator,comprising a) an induction core; and b) a plurality ofsubstantially-identical cells, each cell comprising an F-magnetconfigured to focus a charged particle beam horizontally and defocussaid beam vertically and a D-magnet configured to defocus said beamhorizontally and focus said beam vertically, wherein said cells arearranged in a ring such that the F- and D-magnets alternate, and suchthat the charged particle beam circulates in said ring and is stableafter being injected and until being extracted; wherein the cells aresymmetrically arranged; and wherein the F-magnets and D-magnets havelinear edges; and wherein the F-magnets and D-magnets are configured togenerate a guide magnetic field which varies non-linearly with radiusand to determine reference orbits which close geometrically.
 2. Theaccelerator of claim 1, wherein, for each D-magnet and for eachF-magnet, an edge angle of that magnet at an extraction radius and anedge angle of that magnet at an injection radius are equal to eachother.
 3. The accelerator of claim 1, wherein, for each D-magnet and foreach F-magnet, a magnet length is greater than or equal to 0.01 m. 4.The accelerator of claim 1, wherein, for each D-magnet and for eachF-magnet, an absolute magnetic field strength is less than or equal to 5kG.
 5. The accelerator of claim 1, wherein, for each pair of adjoiningmagnets, a magnet spacing between said magnets at an extraction energyis greater than or equal to 0.05 m.
 6. The accelerator of claim 1,wherein, for each pair of adjoining magnets, a drift length at injectionand a drift length at an energy intermediate between injection andextraction is greater than 0.01 m.
 7. The accelerator of claim 1,wherein: a beam energy is about 0.05 MeV at injection and about 3 MeV atextraction, an average radius is about 0.198 m. at injection and about0.417 m at extraction, a peak field in the F-magnets is about 0.088 kGat injection and about 1.002 kG at extraction, a peak field in theD-magnets is about −0.177 kG at injection and about −0.491 kG atextraction, a magnet length for the F-magnets is about 0.0877 m atinjection and about 0.1348 m at extraction, a magnet length for theD-magnets is about 0.0094 m. at injection and about 0.0976 atextraction, an aperture in the F-magnet is about 0.218 m, an aperture inthe D-magnets is about 0.179 m, there are 8 cells, and a sextupole fieldis sufficient to contain tune variation and permit beam stability. 8.The accelerator of claim 1, wherein: a beam energy is about 0.05 MeV atinjection and about 4 MeV at extraction, an average radius is about0.162 m. at injection and about 0.350 m at extraction, a peak field inthe F-magnets is about 0.082 kG at injection and about 1.105 kG atextraction, a peak field in the D-magnets is about −0.110 kG atinjection and about −1.488 kG at extraction, a magnet length for theF-magnets is about 0.0949 m at injection and about 0.1458 m atextraction, a magnet length for the D-magnets is about 0.0134 m. atinjection and about 0.0294 at extraction, an aperture in the F-magnet isabout 0.190 m, an aperture in the D-magnets is about 0.162 m, and thereare 8 cells.
 9. The accelerator of claim 1, wherein: a beam energy isabout 0.05 MeV at injection and about 3.5 MeV at extraction, an averageradius is about 0.104 m. at injection and about 0.274 m at extraction, apeak field in the F-magnets is about 0.500 kG at injection and about2.034 kG at extraction, a peak field in the D-magnets is about −0.000 kGat injection and about −0.209 kG at extraction, a magnet length for theF-magnets is about 0.0112 m at injection and about 0.061 m atextraction, a magnet length for the D-magnets is about 0.0390 m. atinjection and about 0.0598 at extraction, an aperture in the F-magnet isabout 0.190 m, an aperture in the D-magnets is about 0.161 m, and thereare 8 cells.
 10. The accelerator of claim 1, wherein a beam energy isabout 0.05 MeV at injection and about 9 MeV at extraction, an averageradius is about 0.206 m. at injection and about 0.437 m at extraction, apeak field in the F-magnets is about 0.114 kG at injection and about1.986 kG at extraction, a peak field in the D-magnets is about −0.107 kGat injection and about −2.811 kG at extraction, a magnet length for theF-magnets is about 0.0670 m at injection and about 0.1650 m atextraction, a magnet length for the D-magnets is about 0.0208 m. atinjection and about 0.0398 at extraction, an aperture in the F-magnet isabout 0.230 m, an aperture in the D-magnets is about 0.199 m, and thereare 9 cells.
 11. A method of designing a non-scaling fixed fieldalternating-gradient particle accelerator, wherein there is a non-linearmagnetic field variation with radius, the method comprising a) selectinga first plurality of design parameters for the accelerator; b) selectinga second plurality of equations relating the selected design parameters;c) selecting a third plurality of constraints on at least some of theselected design parameters; d) selecting a fourth plurality ofconstraint equations; e) specifying an intermediate energy; f) selectinga fifth plurality of equations relating to the intermediate energy; g)carrying out an optimizer search to determine at least one potentialdesign for the accelerator; h) verifying at least one potential designby calculating exact orbits and checking tunes for stability over adesired energy range; i) if no potential design has a stable tune,repeating the optimizer search; and j) if at least one potential designhas a stable tune, completing the design of the accelerator based uponsaid potential design.
 12. The method of claim 11, wherein at least someof the second plurality of equations comprise thick lens linear matrixtraces.
 13. The method of claim 11, wherein the first plurality ofdesign parameters comprises: D_(e), a drift distance between an F magnetand a D magnet at extraction; L_(if), L_(ef), L_(id), L_(ed), F magnetand D magnet half-lengths at injection and extraction; B_(if), B_(ef),B_(id), B_(ed), F magnet and D magnet fields at injection andextraction; δx_(if,), a distance from an injection orbit to anextraction orbit in an F magnet; and η_(f), and η_(d), linear edgeangles for F magnets and D-magnets.
 14. The method of claim 11, whereinthe second plurality of equations comprises:${\left. {{{\left. {{{{\left. {{\left. {{{{{\left. {{\left. {{{{{\left. \mspace{20mu} 1 \right)\mspace{14mu} k_{if}L_{if}} + \frac{\theta_{if}}{\rho_{if}} + \frac{\left( {\left( {\theta_{ef} - \theta_{if}} \right) + \eta_{if}} \right)}{\rho_{if}}} = \frac{1}{f_{if}}};}\mspace{20mu} 2} \right)\mspace{14mu} \text{?}}\mspace{20mu} 3} \right)\mspace{14mu} k_{ef}L_{ef}} + \frac{\theta_{ef}}{\rho_{ef}} + \frac{\eta_{ef}}{\rho_{ef}}} = \frac{1}{f_{ef}}};}\mspace{20mu} 4} \right)\mspace{14mu} \text{?}}\mspace{20mu} 5} \right)\mspace{14mu} \theta_{halfcell}} = {{\theta_{if} + \theta_{id}} = {\theta_{ef} + \theta_{ed}}}};}6} \right)\mspace{14mu} {L_{if}\left\lbrack {{\cos \left( \theta_{if} \right)} + {{\sin \left( \theta_{if} \right)}{\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}} \right\rbrack}} = {{L_{ef}{\cos \left( \theta_{ef} \right)}} - {\left\lbrack {{\delta \; x_{if}} - {L_{ef}{\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}}}7} \right)\mspace{14mu} L_{ed}{\cos \left( \theta_{ef} \right)}} = {{L_{id}{\cos \left( \theta_{if} \right)}} + {\delta \; x_{id}{\sin \left( {\theta_{id} + \theta_{if}} \right)}} + {\left\lbrack {{\delta \; x_{if}} + {\left( {L_{if} + D_{i}} \right){\sin \left( \theta_{if} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}$?indicates text missing or illegible when filed
 15. The method of claim11, wherein the third plurality of constraints comprises: 1) magnetlengths less than 0.01 m are excluded; 2) absolute magnetic fieldstrengths greater than 5 kG are excluded; 3) magnet spacings at anextraction energy less than 0.05 m are excluded; and 4) drifts atinjection and intermediate energies less than 0.01 m are excluded. 16.The method of claim 11, wherein the fourth plurality of constraintequations comprises:${\left. {{{\left. {{{\left. {{{\left. {{{\left. \mspace{20mu} 1 \right)\mspace{14mu} {Inj}\; {Radius}} = {{Radi} = {{NSector}\; {{Lihalf}/\pi}}}}\mspace{20mu} 2} \right)\mspace{14mu} {Ext}\; {Radius}} = {{Rade} = {{NSector}\; {{Lehalf}/\pi}}}}\mspace{20mu} 3} \right)\mspace{14mu} \frac{\pi}{NSector}} = {0_{id} + \theta_{if}}}4} \right)\mspace{14mu} \delta \; x_{id}{\cos \left( {\theta_{id} + \theta_{if}} \right)}} = {{\delta \; x_{if}} + {{Lihalf}\; {\sin \left( \theta_{if} \right)}} - {{Lehalf}\; {\sin \left( \theta_{ef} \right)}}}}5} \right)\mspace{14mu} {D_{i}\left\lbrack {{\cos \left( \theta_{if} \right)} - {{\sin \left( \theta_{if} \right)}{\tan \left( \eta_{ed} \right)}}} \right\rbrack}} = {{\left( {L_{ef} + D_{e}} \right){\cos \left( \theta_{ef} \right)}} - {L_{if}{\cos \left( \theta_{if} \right)}} + {\left\lbrack {{\delta \; x_{if}} + {L_{if}{\sin \left( \theta_{if} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}$17. The method of claim 11, wherein the fifth plurality of equationscomprises:${\left. {{{\left. {{{\left. {{{\left. {{{\left. {{{{{\left. {{{{{\left. \mspace{20mu} 1 \right)\mspace{14mu} k_{3\; f}L_{3\; f}} + \frac{\theta_{3\; f}}{\rho_{3\; f}} + \frac{\left( {\left( {\theta_{ef} - \theta_{3\; f}} \right) + \eta_{if}} \right)}{\rho_{3\; f}}} = \frac{1}{f_{3\; f}}};}\mspace{20mu} 2} \right)\mspace{14mu} k_{3\; d}L_{3\; d}} + \frac{\theta_{3\; f} + \eta_{3\; d}}{\rho_{3\; d}}} = \frac{1}{f_{3\; d}}};}\mspace{20mu} 3} \right)\mspace{14mu} \theta_{halfcell}} = {{\theta_{if} + \theta_{id}} = {\theta_{3\; f} + \theta_{3\; d}}}}4} \right)\mspace{14mu} {L_{3\; f}\left\lbrack {{\cos \left( \theta_{3\; f} \right)} + {{\sin \left( \theta_{3\; f} \right)}{\tan \left( {\theta_{ef} + \eta_{ef}} \right)}}} \right\rbrack}} = {{{L_{ef}{\cos \left( \theta_{ef} \right)}} - {\left. \quad{{\left\lbrack {{\delta \; x_{3\; f}} - {L_{ef}{\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( {\theta_{ef} + \eta_{ef}} \right)}};5} \right)\mspace{14mu} L_{ed}{\cos \left( \theta_{ef} \right)}}} = {{L_{3\; d}{\cos \left( \theta_{3\; f} \right)}} + {\delta \; x_{3\; d}{\sin \left( {\theta_{3\; d} + \theta_{3\; f}} \right)}} + {\left\lbrack {{\delta \; x_{3\; f}} + {\left( {L_{3\; f} + D_{3}} \right){\sin \left( \theta_{3\; {if}} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}}}\mspace{20mu} 6} \right)\mspace{14mu} \delta \; x_{3\; d}{\cos \left( {\theta_{3\; d} + \theta_{3\; f}} \right)}} = {{\delta \; x_{3\; f}} + {L\; 3\; {half}\; {\sin \left( \theta_{3\; f} \right)}} - {{Lehalf}\; {\sin \left( \theta_{ef} \right)}}}}7} \right)\mspace{14mu} {D_{3}\left\lbrack {{\cos \left( \theta_{3\; f} \right)} - {{\sin \left( \theta_{3\; f} \right)}{\tan \left( \eta_{ed} \right)}}} \right\rbrack}} = {{\left( {L_{ef} + D_{e}} \right){\cos \left( \theta_{ef} \right)}} - {L_{3\; f}{\cos \left( \theta_{3\; f} \right)}} + {\left\lbrack {{\delta \; x_{3\; f}} + {L_{3\; f}{\sin \left( \theta_{3\; f} \right)}} - {\left( {L_{ef} + D_{e}} \right){\sin \left( \theta_{ef} \right)}}} \right\rbrack {\tan \left( \eta_{ed} \right)}}}}\mspace{20mu} 8} \right)\mspace{14mu} {Rad}\; 3} = {N\; {sector}\; L\; 3\; {{half}/{\pi \mspace{70mu}\left\lbrack {{\delta \; x_{3\; f}} + \left( {l_{ef} + D_{e}} \right)} \right.}}}$18. The method of claim 11, wherein a least squares merit function isused in the optimization.
 19. The method of claim 11, wherein the numberof sectors is between 6 and
 11. 20. A method of designing a non-scalingfixed field alternating-gradient particle accelerator, wherein there isa non-linear magnetic field variation with radius, the method comprisinga) selecting a first plurality of design parameters for the accelerator,a second plurality of equations relating the selected design parameters,a third plurality of constraints on at least some of the selected designparameters, and a fourth plurality of equations relating to anintermediate energy; b) carrying out an optimizer search to determine atleast one potential design for the accelerator; c) verifying at leastone potential design by checking tunes for stability over a desiredenergy range; and d) for at least one potential design with a stabletune, completing the design of the accelerator based upon said potentialdesign.